#8844566

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,4)(6,7)(9,10,15,18)(11,16,12,17)(13,23,24,14)(19,20,21,22)(25,26)(27,28)(29,32)(30,38)(31,34)(33,40)(35,37)(36,39);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,18,17)(13,19,24,21)(14,22,23,20)(25,27,31,32)(26,28,34,29)(30,39,40,35)(33,37,38,36);(1,3,5,8)(2,6,4,7)(11,12)(16,17)(19,21)(20,22)(25,28,31,29)(26,32,34,27)(30,36,40,37)(33,39,38,35);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,18,16)(13,19,24,21)(14,22,23,20)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,20)(18,23)(25,29,31,28)(26,27,34,32)(30,37,40,36)(33,35,38,39);(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,19)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,40)(32,39)(34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8751762);